Question

How many points do you need to graph to get the parabola? How do you determine those points? How does symmetry help us graph?​

Answer 1

Explanation:

the general formula for a parabola is

y = a(x - h)² + k

with (h, k) being the vertex, and "a" being a scaling factor (is sign determines, if the parabola opens upwards or downwards).

so, x and y are point coordinates.

essentially that means we have one equation with 3 variables (h, k, a).

therefore, we need 2 points, if one of them is the vertex (that gives us 2 variables with one stone).

but in general we need 3 points to create and solve 3 equations in the 3 variables.

if we go for the 2- point option, as mentioned, we need the vertex.

y = a(x - h)² + k = ax² - 2ahx + ah² + k

we then name b the factor of x

-2ah = b

and c being the constant term

ah² + k = c

giving us the other form of the parabola equation :

y = f(x) = ax² + bx + c

therefore,

h = -b/2a

k = f(h) = ah² + bh + c

if we go for the general 3-point option, then we just create and solve the 3 equations :

y1 = a(x1 - h)² + k

y2 = a(x2 - h)² + k

y3 = a(x3 - h)² + k

with (x1, y1), (x2, y2), (x3, y3) being the 3 points.

we get the points by selecting any x-value we can think of and calculate the corresponding y via f(x).

the vertex we get in the same way as in the 2-point option. before via the established (h, k) information.

when we know the vertex and one side of the graph (in one side of the vertex), the other side is the exact mirrored image. that's how the symmetry helps us.

Answer 2

You need 2 points to form a parabola. One of those points must be the vertex. The vertex form is y = a(x-h)^2 + k where (h,k) is the vertex.

To find another point on the parabola, plug in a value for x to get its paired y value. Each point is of the form (x,y).

Symmetry is useful because we can mirror one point over the axis of symmetry to another point on the parabola.

For example, the point (1,1) on y = x^2 mirrors over to (-1,1)